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Double Elbow

Explanation of the formula showing coordinates of P2.
点P2の座標を表わす式の詳細な説明図

Coordinates of P2 can be expressed using coordinates of P1 and C.
The upper figure shows only a portion required for explanation
among the figures of Double Elbow.

P1X is a horizontal line. Define the leg of the vertical line taken down from P2 to P1X as point R.
Define the leg of the vertical line which took down the two middle points of P1 and C to P1X from Q and Q to R.
Set the middle point of points P1 and C to Q. Define the leg of the vertical line taken down from Q to P1X as T.
If the point of extending line P2C in the direction of C, and crossing P1X is set to Z, since triangle P1P2Z is bi-equilateral triangle, it leads to P1R=RZ, and line QR is parallel to line P2Z. Therefore, angle TRQ are 30degs.
Coordinate X of point P2 is coordinates X of point Q plus TR.
TR=TQ/tan30deg.
Point Q is the middle points of P1 and C, therefore the coordinate X of P2 is as shown on previous page.
It is the same also about coordinate Y.
The above was in the case of P2. Also in P3, it is completely the same.
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